Bending loss is known to limit the performance of optical waveguides when configurations with sharp bends or small radii are required. Single mode waveguides, used extensively in integrated optics applications, are particularly susceptible to bending loss.
Bending loss occurs when the mode travelling in a curved waveguide couples to radiation modes outside the waveguide, resulting in a loss of energy from the waveguide. Coupling occurs when the phase velocity of the energy travelling in the bend matches the phase velocity outside the waveguide. FIG. 1a illustrates a straight section of waveguide 1 and shows that a wavefront phase velocity (v) of a guided mode is equal across the waveguide. As shown in FIG. 1b for a curved waveguide 2 the phase velocity of the guided mode is proportional to the radius of curvature (R.sub.1 or R.sub.2 each measured to a point on the surface of the substrate) in that the outer-most portion of the mode must travel faster (v.sub.o) than the inner-most portion of the mode (v.sub.i) in order to preserve the phase front. The speed of the waveguide mode goes inversely with the effective index of refraction (n.sub.eff).
FIG. 2a and FIG. 2b illustrate the actual waveguide index of refraction profiles corresponding to the waveguides of FIG. 1a and FIG. 1b, respectively, and FIGS. 3a and 3b illustrate the effective index of refraction profile for the waveguides of FIG. 1a, FIG. 1b and FIG. 2a, FIG. 2b, respectively. As can be seen, the effective index of refraction varies asymmetrically across the curved waveguide from a value of n.sub.1 at R.sub.1 to a substantially lower value of n.sub.2 at R.sub.2, it being noted that the value of n.sub.2 approaches the value of the index of refraction in the surrounding material (n.sub.o) for sufficiently large values of R.sub.2 and small values of delta n. When n.sub.2 equals n.sub.0, the guided mode is phase velocity matched to the substrate radiation modes and bending loss occurs.
That is, if the radius is sufficiently sharp v.sub.o becomes equal to the velocity in the material outside of the waveguide 2 and an undesirable coupling of energy out of the waveguide 2 occurs. The amount of energy coupled out of the waveguide per unit length is proportional to the energy in the mode outside the waveguide.
It is therefore an object of the invention to provide an optic waveguide structure having a curved geometry that does not couple a significant portion of the mode energy propagating therethrough to material outside of the waveguide.
It is another object of the invention to provide a method for fabricating an optic waveguide structure having a curved geometry such that the waveguide structure does not couple a significant portion of the mode energy propagating therethrough to material outside of the waveguide.